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Sampling and series expansion for linear canonical transform

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Abstract

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This paper investigates new sampling relations in the LCT domain. Firstly, the relationship between linear canonical series (LCS) and LCT is introduced. The LCS expansion coefficients are the sampled values of LCT. Then, based on the conventional Fourier series and LCS, two new sampling relations in the LCT domain are presented, where the signal in the time domain is reconstructed from the samples of its LCT directly. The first theorem considers signals band-limited in some LCT domain, and the second deals with signals band-limited in the conventional Fourier transform domain.

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References

  1. Moshinsky, M., Quesne, C.: Linear canonical transformations and their unitary representations. J. Math. Phys. 12, 1772–1783 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  2. Wolf, K.B.: Integral Transforms in Science and Engineering, Chap. 9. Plenum, New York (1979)

  3. Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2000)

    Google Scholar 

  4. Pei, S.C., Ding, J.J.: Relations between fractional operations and time-frequency distributions, and their applications. IEEE Trans. Signal Process. 49, 1638–1655 (2001)

    Article  MathSciNet  Google Scholar 

  5. Stern, A.: Why is the linear canonical transform so little known? Proc. AIP 860, 225–234 (2006)

    Article  Google Scholar 

  6. Alieva, T., Bastiaans, M.J.: Properties of the linear canonical integral transformation. J. Opt. Soc. Am. A 24, 3658–3665 (2007)

    Article  MathSciNet  Google Scholar 

  7. Erseghe, T., Laurenti, N., Cellini, V.: A multicarrier architecture based upon the affine Fourier transform. IEEE Trans. Commun. 53(5), 853–862 (May 2005)

    Google Scholar 

  8. Onural, L., Gotchev, A., Ozaktas, H.M., Stoykova, E.: A survey of signal processing problems and tools in holographic three-dimensional television. IEEE Trans. Circuit Syst. Video Technol. 17, 1631–1646 (2007)

    Article  Google Scholar 

  9. Sharma, K.K., Joshi, S.D.: signal separation using linear canonical and fractional Fourier transform. Opt. Commun. 265, 454–460 (2006)

    Article  Google Scholar 

  10. Barshan, B., Kutay, M.A., Ozaktas, H.M.: Optimal filtering with linear canonical transformations. Opt. Commun. 135, 32–36 (1997)

    Article  Google Scholar 

  11. Stern, A.: Uncertainty principles in linear canonical transform domains and some of their implications in optics. J. Opt. Soc. Am. A. 25(3), 647–652 (2008)

    Article  Google Scholar 

  12. Shi, J., Liu, X., Zhang, N.: On uncertainty principles for linear canonical transform of complex signals via operator methods. Signal Image Video Process. 8, 85–93 (2014)

    Article  Google Scholar 

  13. Wei, D., Ran, Q., Li, Y.: A convolution and product theorem for the linear canonical transform. IEEE Signal Process. Lett. 16, 853–856 (2009)

    Article  Google Scholar 

  14. Wei, D., Ran, Q., Li, Y.: A convolution and correlation theorem for the linear canonical transform and its application. Circuits Syst. Signal Process. 31, 301–312 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Goel, N., Singh, K.: Modified correlation theorem for the linear canonical transform with representation transformation in quantum mechanics. Signal Image Video Process. 8, 595–601 (2014)

    Article  Google Scholar 

  16. Shinde, S.: Two channel paraunitary filter banks based on linear canonical transform. IEEE Trans. Signal Process. 59(2), 832–836 (2011)

    Article  MathSciNet  Google Scholar 

  17. Shi, J., Liu, X., Zhang, N.: Generalized convolution and product theorems associated with linear canonical transform. Signal Image. Video Process. (2012). doi:10.1007/s11760-012-0348-7

  18. Xiang, Q., Oin, K.Y.: Convolution, correlation, and sampling theorems for the offset linear canonical transform. Signal Image Video Process. 8, 433–442 (2014)

    Article  Google Scholar 

  19. Marks II, R.J.: Advanced Topics in Shannon Sampling and Interpolation Theory. Springer, Berlin (1993)

    Book  Google Scholar 

  20. Oppenheim, A.V., Schafer, R.W.: Digital Signal Processing. Prentice Hall, India (1994)

  21. Stern, A.: Sampling of linear canonical transformed signals. Signal Process. 86, 1421–1425 (2006)

    Article  MATH  Google Scholar 

  22. Stern, A.: Sampling of compact signals in offset linear canonical transform domains. Signal Image Video Process. 1(4), 359–367 (2007)

    Article  MATH  Google Scholar 

  23. Li, B.Z., Tao, R., Wang, Y.: New sampling formulae related to linear canonical transform. Signal Process. 87, 983–990 (2007)

    Article  MATH  Google Scholar 

  24. Li, B.Z., Xu, T.Z.: Spectral analysis of sampled signals in the linear canonical transform domain. Math. Probl. Eng. (2012), Article ID 536464, 11 pp

  25. Zhao, H., Ran, Q.W., Ma, J., Tan, L.Y.: On bandlimited signals associated with linear canonical transform. IEEE Signal Process. Lett. 16, 343–345 (2009)

    Article  Google Scholar 

  26. Healy, J.J., Sheridan, J.T.: Sampling and discretization of the linear canonical transform. Signal Process. 89, 641–648 (2009)

    Article  MATH  Google Scholar 

  27. Healy, J.J., Sheridan, J.T.: Cases where the linear canonical transform of a signal has compact support or is band-limited. Opt. Lett. 33, 228–230 (2008)

    Article  Google Scholar 

  28. Wei, D., Ran, Q., Li, Y.: Sampling of bandlimited signals in the linear canonical transform domain. Signal Image Video Process. 7, 553–558 (2013)

    Article  Google Scholar 

  29. Tao, R., Li, B.Z., Wang, Y., Aggrey, G.K.: On sampling of band-limited signals associated with the linear canonical transform. IEEE Trans. Signal Process. 56(11), 5454–5464 (2008)

    Article  MathSciNet  Google Scholar 

  30. Sharma, K.K.: Approximate signal reconstruction using nonuniform samples in fractional Fourier and linear canonical transform domains. IEEE Trans. Signal Process. 53, 4573–4578 (2009)

    Article  Google Scholar 

  31. Zhao, H., Ran, Q.W., Tan, L.Y., Ma, J.: Reconstruction of bandlimited signals in linear canonical transform domain from finite nonuniformly spaced samples. IEEE Signal Process. Lett. 16(12), 1047–1050 (2009)

    Article  Google Scholar 

  32. Li, C.P., Li, B.Z., Xu, T.Z.: Approximating bandlimited signals associated with the LCT domain from nonuniform samples at unknown locations. Signal Process. 92, 1658–1664 (2012)

    Article  Google Scholar 

  33. Wei, D., Ran, Q., Li, Y.: Multichannel sampling and reconstruction of bandlimited signals in the linear canonical transform domain. IET Signal Process. 5(8), 717–727 (2011)

    Google Scholar 

  34. Wei, D., Ran, Q., Li, Y.: Reconstruction of band-limited signals from multichannel and periodic nonuniform samples in the linear canonical transform domain. Opt. Commun. 284(19), 4307–4315 (2011)

    Article  Google Scholar 

  35. Wei, D., Ran, Q., Li, Y.: Multichannel sampling expansion in the linear canonical transform domain and its application to superresolution. Opt. Commun. 284(23), 5424–5429 (2011)

    Article  Google Scholar 

  36. Wei, D., Li, Y.: Reconstruction of multidimensional bandlimited signals from multichannel samples in the linear canonical transform domain. IET Signal Process. (2013). doi:10.1049/iet-spr.2013.0240

  37. Wei, D., Li, Y.: Sampling reconstruction of N-dimensional bandlimited images after multilinear filtering in fractional Fourier domain. Opt. Commun. 295, 26–35 (2013)

    Article  Google Scholar 

  38. Pei, S.C., Yeh, M.H., Luo, T.L.: Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform. IEEE Trans. Signal Process. 47, 2883–2888 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  39. Oppenheim, A.V., Willsky, A.S.: Signals and System. Prentice-Hall, Englewood Cliffs (1983)

  40. Papoulis, A.: Signal Analysis. McGraw-Hill, New York (1977)

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Acknowledgments

The authors would like to thank the editor and all the anonymous referees for their valuable comments and suggestions that improved the clarity and quality of this manuscript. This work was supported by National Natural Science Foundation of China under Grant 61301283 and also sponsored by the Fundamental Research Funds for the Central Universities under Grants K5051370011, BDY111407, K5051370024 and K5051370006.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    Deyun Wei & Yuan-Min Li

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  1. Deyun Wei
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  2. Yuan-Min Li
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Correspondence to Deyun Wei.

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Wei, D., Li, YM. Sampling and series expansion for linear canonical transform. SIViP 8, 1095–1101 (2014). https://doi.org/10.1007/s11760-014-0638-3

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  • DOI: https://doi.org/10.1007/s11760-014-0638-3

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